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A Discovery in Architecture: 15th Century Islamic Architecture Presages 20th Century Mathematics

This article relates the discovery by two American scholars, Paul J. Steinhardt and Peter J. Lu (respectively from the department of physics at Princeton and Harvard universities) that medieval Islamic artists produced intricate decorative patterns using geometrical techniques that were not understood by Western mathematics until the 20th century.

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Figure 1. Archway from the Darb-i Imam shrine in Isfahan, Iran, which was built in 1453 C.E. The larger pentagons outlined in pale blue were constructed using a large-scale girih tile pattern, and the small white pentagons were constructed using a small-scale girih tile pattern (Image courtesy of K. Dudley and M. Elliff).

"Sophisticated Geometry in Islamic Architecture", "Geometry meets artistry in medieval tile work", "Geometry Meets Arts in Islamic Tiles". These are some of the sentences that highlight articles we can find this week in the news agencies science dispatches and in the media coverage of an exciting discovery published in the last issue of Science Magazine: Peter J. Lu and Paul J. Steinhardt "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture," Science vol. 315, n° 1106 (2007).

The article relates the discovery by two American scholars, Paul J. Steinhardt and Peter J. Lu (respectively from the department of physics at Princeton and Harvard universities) that medieval Islamic artists produced intricate decorative patterns using geometrical techniques that were not understood by Western mathematics until the 20th century. The combinations of ornate stars and polygons that have adorned mosques and palaces since the 15th century were created using a set of just five template tiles, which could generate patterns with a kind of symmetry that eluded formal mathematical description for another 500 years.

The discovery suggests that the Islamic artisans who created these typical girih designs had an intuitive understanding of highly complex mathematical concepts. "We can't say for sure what it means," said Peter J. Lu, a PhD student in physics, quoted by The Times in February 23, 2007. "It could be proof of a major role of mathematics in medieval Islamic art or it could have been just a way for artisans to construct their art more easily. It would be incredible if it were all coincidence. At the very least, it shows us a culture that we often don't credit enough was far more advanced than we thought before."

Girih designs feature arrays of tessellating polygons of multiple shapes, and are often overlaid with a zigzag network of lines. It had been assumed that straightedge rulers and compasses were used to create them — an exceptionally difficult process as each shape must be precisely drawn. From the 15th century, however, some of these designs are symmetrical in a way known today as "quasicrystalline". Such forms have either fivefold or tenfold rotational symmetry — meaning they can be rotated to either five or ten positions that look the same — and their patterns can be infinitely extended without repetition. The principles behind quasicrystalline symmetry were calculated by the Oxford mathematician Roger Penrose in the 1970s, but it is now clear that Islamic artists were creating them more than 500 years earlier.

Figure 2. A 15th-century Timurid-Turkmen scroll now held by the Topkapi Palace Museum in Istanbul. The faint reddish lines outline the shapes of the underlying tiles. One example of each shape has been shaded in the picture (Image from www.sciencenews.org).

Peter Lu, one of the authors of this discovery, began wondering whether there were quasicrystalline forms in Islamic art after seeing decagonal artworks in Uzbekistan, while he was there for professional reasons. On returning to Harvard, he started searching the university's vast library of Islamic art for quasicrystalline designs. He found several, as well as architectural scrolls that contained the outlines of five polygon templates — a ten-sided decagon, a hexagon, a pentagon, a rhombus and a bow-tie shape — that can be combined and overlaid to create such patterns.

In keeping with the Islamic tradition of not depicting images of people or animals, many religious buildings were decorated with geometric star-and-polygon patterns, often overlaid with a zigzag network of lines. Lu and Steinhardt show in their study published in the journal Science that by the 13th century Islamic artisans had begun producing patterns using a small set of decorated, polygonal tiles which they call "girih" tiles.

Art historians have until now assumed that the intricate tile work had been created using straight edges and compasses, but the study suggests the Islamic artisans were using a basic toolkit of girih tiles made up of shapes such as the decagon, pentagon, diamond and hexagon.

Straight edges and compasses work fine for the recurring symmetries of the simplest patterns we see, but it probably required far more powerful tools to fully explain the elaborate tiling with decagonal [10-sided] symmetry," Mr Lu said, quoted by The Independent on 25 February 2007. He adds that "individually placing and drafting hundreds of decagons with a straight edge would have been exceedingly cumbersome. It's more likely these artisans used particular tiles that we've found by decomposing the artwork".

The scientists found that by 1453, Islamic architects had created overlapping patterns with girih tiles at two sites to produce near-perfect quasicrystalline patterns that did not repeat themselves. "The fact that we can explain so many sets of tiling, from such a wide range of architectural structures throughout the Islamic world with the same set of tiles, makes this an incredibly interesting universal picture," Mr Lu said.

We are glad at Muslimheritage.com to review this discovery and to provide links to the article published in Science and to supporting online material, as well as links to media coverage resources for the interested readers.

As a background to the present day research on Islamic architecture as a conjunction of mathematics, arts and practical knowledge, we can mention the following project: Muqarnas is a project concerned with the the mathematical basis of the Muqarnas, which is the Arabic word for the 3-D design used to decorate domes, ceilings and walls, known to the Europeans as "stalactite decoration". It is an architectural ornament developed around the middle of the tenth century in north eastern Iran and almost simultaneously, but apparently independently, in North Africa [1]. The project was led by scholars from Heidelberg University in Germany: http://www.iwr.uni-heidelberg.de/groups/ngg/Muqarnas/index.php?L=E, under the direction of Yvonne Dold-Samplonius. The group included Yvonne Dold-Samplonius, Silvia Harmsen, Susanne Krömker and Michael Winckler (Heidelberg University, Interdisciplinary Center for
Scientific Computing of the Ruprecht-Karls-University of Heidelberg) and several international cooperation Partners: Gülrü Necipoglu Sackler (Museum Aga Khan Chair for the History of Architecture, Harvard), Mohammad Al-Assad (Center for the Study of the Built Environment, Amman), and Jan P. Hogendijk (Mathematical Institute, University of Utrecht) [2].

In the same period, Alpay Özdural, a scholar from the Eastern Mediterranean University in North Cyprus conducted research the results of which he published in a series of articles [3]. Unfortunately the unexpected passing away by a heart attack on 22 February 2003 of this brilliant scholar has put an end to this wave of promising research.

[2] An outcome of earlier work by this group of scholars concerning al-Kashi's contribution to architecture was issued in 1996 as a video tape distributed by The American Mathematical Society: Qubba for al-Kâshî. Video Tape(Heidelberg: Institut für Wissenschaftliches Rechnen, Universität Heidelberg).

[3]Alpay Özdural 1995. "Omar Khayyâm, Mathematicians and Conversazioni with Artisans." Journal of the Society of Architectural Historians vol 54: pp. 54-71 (etablishes a connection between a triangle constructed by al-Khayyâm in his treatise of algebra and mosaic drawings);

Alpay Özdural 1996. "On Interlocking Similar or Corresponding Figures and Ornamental Patterns of Cubic Equations." Muqarnas (Leiden) vol. 13: pp. 191-211 (a partial analysis of an Iranian MS from the 16th century including mosaic drawings that could not be drawn by compas and ruler);